Question: $ 25^{-\frac{7}{2}}$
$= \left(\dfrac{1}{25}\right)^{\frac{7}{2}}$ $= \left(\left(\dfrac{1}{25}\right)^{\frac{1}{2}}\right)^{7}$ To simplify $\left(\dfrac{1}{25}\right)^{\frac{1}{2}}$ , figure out what goes in the blank: $\left(? \right)^{2}=\dfrac{1}{25}$ To simplify $\left(\dfrac{1}{25}\right)^{\frac{1}{2}}$ , figure out what goes in the blank: $\left({\dfrac{1}{5}}\right)^{2}=\dfrac{1}{25}$ so $ \left(\dfrac{1}{25}\right)^{\frac{1}{2}}=\dfrac{1}{5}$ So $\left(\dfrac{1}{25}\right)^{\frac{7}{2}}=\left(\left(\dfrac{1}{25}\right)^{\frac{1}{2}}\right)^{7}=\left(\dfrac{1}{5}\right)^{7}$ $= \left(\dfrac{1}{5}\right)^{7}$ $= \left(\dfrac{1}{5}\right)\cdot\left(\dfrac{1}{5}\right)\cdot \left(\dfrac{1}{5}\right)\cdot \left(\dfrac{1}{5}\right)\cdot \left(\dfrac{1}{5}\right)\cdot \left(\dfrac{1}{5}\right)\cdot \left(\dfrac{1}{5}\right)$ $= \dfrac{1}{25}\cdot\left(\dfrac{1}{5}\right)\cdot \left(\dfrac{1}{5}\right)\cdot \left(\dfrac{1}{5}\right)\cdot \left(\dfrac{1}{5}\right)\cdot \left(\dfrac{1}{5}\right)$ $= \dfrac{1}{125}\cdot\left(\dfrac{1}{5}\right)\cdot \left(\dfrac{1}{5}\right)\cdot \left(\dfrac{1}{5}\right)\cdot \left(\dfrac{1}{5}\right)$ $= \dfrac{1}{625}\cdot\left(\dfrac{1}{5}\right)\cdot \left(\dfrac{1}{5}\right)\cdot \left(\dfrac{1}{5}\right)$ $= \dfrac{1}{3125}\cdot\left(\dfrac{1}{5}\right)\cdot \left(\dfrac{1}{5}\right)$ $= \dfrac{1}{15625}\cdot\left(\dfrac{1}{5}\right)$ $= \dfrac{1}{78125}$